Linear Latency Research Articles

Transportation network with externalities

In a transportation network comprised of parallel routes with linear latency functions, we study how externalities among different routes affect the socially optimal allocation and the equilibrium allocation of traffic flows. Assuming that the externalities are not too severe, we analytically derive a system of equations that define the optimal distribution of the traffic flow with minimum social cost. We also solve for the Wardrop equilibrium without route charges in which every commuter minimizes the travel time. Furthermore, we characterize the Wardrop equilibrium with route charges in which two carriers set prices to maximize their own profits and commuters choose which carrier’s service to use in order to minimize the sum of monetary cost (service price paid) and waiting cost (travel time spent). Our results show that Wardrop’s first principle (user equilibrium) remains valid with mild externality level, and his second principle (social optimum), which usually fails to hold in equilibrium in a standard setup without externalities, can be achieved together with the first principle under some externality conditions, leading to the price of anarchy equal to 1. We also show that when the traffic flow goes to infinity, the price of anarchy can still be greater than 1 with the existence of externalities.

Shortest Paths from a Group Perspective—A Note on Selfish Routing Games with Cognitive Agents

This paper presents an analysis of the effects of cognitive agents employing selfish routing behavior in traffic networks with linear latency functions. Selfish routing occurs when each agent traveling on a network acts in a purely selfish manner, therefore the Braess Paradox is likely to occur. The Braess Paradox describes a situation where an additional edge with positive capacity is added to a given network, which leads to higher total system delay. By applying the concept of cognitive agents, each agent is able to make a range of non-selfish and selfish decisions. In addition, each agent has to cope with uncertainty in terms of travel time information associated with the traffic system, a factor in real-world traffic networks. This paper evaluates the influence of travel time uncertainty, and possible non-selfish decisions of the agents on overall network delay. The results indicate that both non-selfish behavior and uncertainty have an influence on overall travel delay. In addition, understanding the influence of cognitive agents on delay can help to better plan and influence traffic flows resulting in “closer to optimal” flows involving overall lower delays.

Non-atomic one-round walks in congestion games

In this paper we study the approximation ratio of the solutions achieved after an ϵ-approximate one-round walk in non-atomic congestion games. Prior to this work, the solution concept of one-round walks had been studied for atomic congestion games with linear latency functions only (Christodoulou et al. [1], Bilò et al. [2]). We give an explicit formula to determine the approximation ratio for non-atomic congestion games having general latency functions. In particular, we focus on polynomial latency functions, and, we prove that the approximation ratio is exactly ((1+ϵ)(p+1))p+1 for every polynomial of degree p. Then, we show that, by resorting to static (resp. dynamic) resource taxation, the approximation ratio can be lowered to (1+ϵ)p+1(p+1)p (resp. (1+ϵ)p+1(p+1)!).

Achieving target equilibria in network routing games without knowing the latency functions

The analysis of network routing games typically assumes precise, detailed information about the latency functions. Such information may, however, be unavailable or difficult to obtain. Moreover, one is often primarily interested in enforcing a desired target flow as an equilibrium. We ask whether one can achieve target flows as equilibria without knowing the underlying latency functions. We give a crisp positive answer to this question. We show that one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an oracle that takes tolls as input and outputs the resulting equilibrium flow. This result is obtained via a novel application of the ellipsoid method, and extends to various other settings. We obtain improved query-complexity bounds for series-parallel networks, and single-commodity routing games with linear latency functions. Our techniques provide new insights into network routing games.

Analysis of Price of Total Anarchy in Congestion Games via Smoothness Arguments

Efficiency loss may exist at every stage of repeated play. With the rapid development of our society, how to improve the overall efficiency in repeated games becomes increasingly crucial. This paper studies the performance of a sequence of action profiles generated by repeated play in a multiple origin-destination network. To analyze the overall efficiency of these action profiles, the price of total anarchy (POTA), defined as the worst-case ratio of the average total latency over a period of time and the total latency of any optimal strategy, is adopted. First of all, it is shown that the sequence of action profiles generated by best response principle with inertia possesses almost sure no-regret property. Then, we identify the upper bound of POTA via smoothness arguments for both linear and nonlinear latency cases. After that, we analyze the effect of road pricing on POTA in traffic networks with linear latency function. It is shown that the road price we designed can reduce the upper bound of POTA compared with that without road price. In addition, how the inaccurate parameter information of the latency function affects the POTA is discussed for the linear latency case. Moreover, in a traffic network with heterogeneous players, we show that the upper bound of POTA is the same as that in a network with homogeneous players as time goes to infinity. Finally, the results are applied to a traffic routing problem based on the real traffic data of Singapore.

Approximate Pure Nash Equilibria in Weighted Congestion Games

We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ --approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ . Do such equilibria exist for small values of ρ ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ -games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5/2 + O γ for arbitrarily small γ > 0; for latency functions with maximum degree d ≥ 2, it is d 2 d + o ( d ). The running time is polynomial in the number of bits in the representation of the game and 1/ γ . As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO ( d 2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ -games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications.

The Strong Price of Anarchy of Linear Bottleneck Congestion Games

We study the inefficiency of equilibrium outcomes in Bottleneck Congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We analyze the (strong) Price of Anarchy of these games for a natural load balancing social cost objective, i.e., minimize the maximum latency of a facility. In our studies, we focus on Bottleneck Congestion games with linear latency functions. These games still constitute a rich class of games and generalize, for example, Load Balancing games with identical or uniformly related machines (with or without restricted assignments). We derive upper and (asymptotically) matching lower bounds on the (strong) Price of Anarchy of these games. We also derive more refined bounds for several special cases of these games, including the cases of identical player weights, identical latency functions and symmetric strategy sets. Further, we provide lower bounds on the Price of Anarchy for k-strong equilibria.

Stability vs. optimality in selfish ring routing

We study asymmetric atomic selfish routing in ring networks, which has diverse practical applications in network design and analysis. We are concerned with minimizing the maximum latency of source-destination node-pairs over links with linear latencies. We show that there exists an optimal solution that is a 9-approximate Nash equilibrium, significantly improving the existing upper bound of 54 on the instability factor. We present fast implementation of the best response dynamics for computing a Nash equilibrium. Furthermore, we perform empirical study on the price of stability, narrowing the gap between the lower and upper bounds to 0.7436.

On the hardness of network design for bottleneck routing games

In routing games, the selfish behavior of the players may lead to a degradation of the network performance at equilibrium. In more than a few cases however, the equilibrium performance can be significantly improved if we remove some edges from the network. This counterintuitive fact, widely known as Braessʼs paradox, gives rise to the (selfish) network design problem, where we seek to recognize routing games suffering from the paradox, and to improve their equilibrium performance by edge removal. In this work, we investigate the computational complexity and the approximability of the network design problem for non-atomic bottleneck routing games, where the individual cost of each player is the bottleneck cost of her path, and the social cost is the bottleneck cost of the network, i.e. the maximum latency of a used edge. We first show that bottleneck routing games do not suffer from Braessʼs paradox either if the network is series-parallel, or if we consider only subpath-optimal Nash flows. On the negative side, we prove that even for games with strictly increasing linear latencies, it is NP-hard not only to recognize instances suffering from the paradox, but also to distinguish between instances for which the Price of Anarchy (PoA) can decrease to 1 and instances for which the PoA cannot be improved by edge removal, even if their PoA is as large as Ω(n0.121). This implies that the network design problem for linear bottleneck routing games is NP-hard to approximate within a factor of O(n0.121−ε), for any constant ε>0. The proof is based on a recursive construction of hard instances that carefully exploits the properties of bottleneck routing games, and may be of independent interest. On the positive side, we present an algorithm for finding a subnetwork that is almost optimal with respect to the bottleneck cost of its worst Nash flow, when the worst Nash flow in the best subnetwork routes a non-negligible amount of flow on all used edges. We show that the running time is essentially determined by the total number of paths in the network, and is quasipolynomial when the number of paths is quasipolynomial.

Braess's paradox in expanders

AbstractExpander graphs are known to facilitate effective routing and most real‐world networks have expansion properties. At the other extreme, it has been shown that in some special graphs, removing certain edges can lead to moreefficient routing. This phenomenon is known as Braess's paradox and is usually regarded as a rare event. In contrast to what one might expect, we show that Braess's paradox is ubiquitous in expander graphs. Specifically, we prove that Braess's paradox occurs in a large class of expander graphs with continuous convex latency functions. Our results extend previous work which held only when the graph was both denser and random and for random linear latency functions. We identify deterministic sufficient conditions for a graph with as few as a linear number of edges, such that Braess's Paradox almost always occurs, with respect to a general family of random latency functions. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Efficient methods for selfish network design

Intuitively, Braess’s paradox states that destroying a part of a network may improve the common latency of selfish flows at Nash equilibrium. Such a paradox is a pervasive phenomenon in real-world networks. Any administrator who wants to improve equilibrium delays in selfish networks, is facing some basic questions:–Is the network paradox-ridden?–How can we delete some edges to optimize equilibrium flow delays?–How can we modify edge latencies to optimize equilibrium flow delays? Unfortunately, such questions lead to NP-hard problems in general. In this work, we impose some natural restrictions on our networks, e.g. we assume strictly increasing linear latencies. Our target is to formulate efficient algorithms for the three questions above. We manage to provide: –A polynomial-time algorithm that decides if a network is paradox-ridden, when latencies are linear and strictly increasing.–A reduction of the problem of deciding if a network with (arbitrary) linear latencies is paradox-ridden to the problem of generating all optimal basic feasible solutions of a Linear Program that describes the optimal traffic allocations to the edges with constant latency.–An algorithm for finding a subnetwork that is almost optimal wrt equilibrium latency. Our algorithm is subexponential when the number of paths is polynomial and each path is of polylogarithmic length.–A polynomial-time algorithm for the problem of finding the best subnetwork which outperforms any known approximation for the case of strictly increasing linear latencies.–A polynomial-time method that turns the optimal flow into a Nash flow by deleting the edges not used by the optimal flow, and performing minimal modifications on the latencies of the remaining ones. Our results provide a deeper understanding of the computational complexity of recognizing the most severe manifestations of Braess’s paradox, and our techniques show novel ways of using the probabilistic method and of exploiting convex separable quadratic programs.

A unified spatio-temporal parallelization framework for accelerated Monte Carlo radiobiological modeling of electron tracks and subsequent radiation chemistry

Monte Carlo (MC) nano-scale modeling of the cellular damage is desirable but most times is prohibitive for large scaled systems due to their intensive computational cost. In this study a parallelized computational framework is presented, for accelerated MC simulations of both particle propagation and subsequent radiation chemistry at the subcellular level. Given the inherent parallelism of the electron tracks, the physical stage was “embarrassingly parallelized” into a number of independent tasks. For the chemical stage, the diffusion–reaction of the radical species was simulated with a time-driven kinetic Monte Carlo algorithm (KMC) based on the Smoluchowski formalism and the parallelization was realized by employing a spatio-temporal linked-list cell method based on a spatial subdivision with a uniform grid. The evaluation of our method was established on two metrics: speedup and efficiency. The results indicated a linear speedup ratio for the physical stage and a linear latency for shared- versus a distributed-memory system with a maximum of 3.6⋅10−3% per electron track. For the chemical stage, a series of simulations were performed to show how the execution time per step was scaling with respect to the number of radical species and a 5.7× speedup was achieved when a larger number of reactants were simulated and eight processors were employed. The simulations were deployed on the Amazon EC2 infrastructure. It is also elucidated how the overhead started becoming significant as the number of reactant species decrease relative to the number of processors. The method reported here lays the methodological foundations for accelerated MC simulations and allows envisaging a future use for large-scale radiobiological modeling of multi-cellular systems involved into a clinical scenario.

Pairwise cooperations in selfish ring routing for minimax linear latency

This paper studies the selfish routing game in ring networks with a load-dependent linear latency on each link. We adopt the asymmetric atomic routing model. Each player selfishly chooses a route to connect his source-destination pair, aiming at the lowest latency of his route, while the system objective is to minimize the maximum latency among all routes of players. The effectiveness of these routing games is often measured by the price of anarchy (PoA), the worst-case ratio between the maximum latencies in a Nash equilibrium (NE) and in a system optimum, where NE refers to a “stable state” among all players, from which no player has the incentive to deviate unilaterally. In classical setting, no cooperation is allowed and 16 stands as the current best upper bound on the PoA of such selfish ring routing. In this paper we show that the PoA is at most 10.16 provided cooperations within pairs of players are allowed, where any two players could change their routes simultaneously if neither would experience a longer latency and at least one would experience a shorter latency.

Routing (un-) splittable flow in games with player-specific affine latency functions

In this work we study weighted network congestion games with player-specific latency functions where selfish players wish to route their traffic through a shared network. We consider both the case of splittable and unsplittable traffic. Our main findings are as follows. For routing games on parallel links with linear latency functions, we introduce two new potential functions for unsplittable and for splittable traffic, respectively. We use these functions to derive results on the convergence to pure Nash equilibria and the computation of equilibria. For several generalizations of these routing games, we show that such potential functions do not exist. We prove tight upper and lower bounds on the price of anarchy for games with polynomial latency functions. All our results on the price of anarchy translate to general congestion games.

Competitive routing over time

Congestion games are a fundamental and widely studied model for selfish allocation problems like routing and load balancing. An intrinsic property of these games is that players allocate resources simultaneously and instantly. This is particularly unrealistic for many network routing scenarios, which are one of the prominent application scenarios of congestion games. In many networks, load travels along routes over time and allocation of edges happens sequentially. In this paper, we consider two frameworks that enhance network congestion games with a notion of time. We introduce temporal network congestion games that are based on coordination mechanisms — local policies that allow to sequentialize traffic on the edges. In addition, we consider congestion games with time-dependent costs, in which travel times are fixed but quality of service of transmission varies with load over time. We study existence and complexity properties of pure Nash equilibria and best-response strategies in both frameworks for the special case of linear latency functions. In some cases our results can be used to characterize convergence properties of various improvement dynamics, by which the population of players can reach equilibrium in a distributed fashion.

Taxes for linear atomic congestion games

We study congestion games where players aim to access a set of resources. Each player has a set of possible strategies and each resource has a function associating the latency it incurs to the players using it. Players are non--cooperative and each wishes to follow a strategy that minimizes her own latency with no regard to the global optimum. Previous work has studied the impact of this selfish behavior on system performance. In this article, we study the question of how much the performance can be improved if players are forced to pay taxes for using resources. Our objective is to extend the original game so that selfish behavior does not deteriorate performance. We consider atomic congestion games with linear latency functions and present both negative and positive results. Our negative results show that optimal system performance cannot be achieved even in very simple games. On the positive side, we show that there are ways to assign taxes that can improve the performance of linear congestion games by forcing players to follow strategies where the total latency suffered is within a factor of 2 of the minimum possible; this result is shown to be tight. Furthermore, even in cases where in the absence of taxes the system behavior may be very poor, we show that the total disutility of players (latency plus taxes) is not much larger than the optimal total latency. Besides existential results, we show how to compute taxes in time polynomial in the size of the game by solving convex quadratic programs. Similar questions have been extensively studied in the model of non-atomic congestion games. To the best of our knowledge, this is the first study of the efficiency of taxes in atomic congestion games.

On the Performance of Approximate Equilibria in Congestion Games

We study the performance of approximate Nash equilibria for congestion games with polynomial latency functions. We consider how much the price of anarchy worsens and how much the price of stability improves as a function of the approximation factor e. We give tight bounds for the price of anarchy of atomic and non-atomic congestion games and for the price of stability of non-atomic congestion games. For the price of stability of atomic congestion games we give non-tight bounds for linear latencies. Our results not only encompass and generalize the existing results of exact equilibria to e-Nash equilibria, but they also provide a unified approach which reveals the common threads of the atomic and non-atomic price of anarchy results. By expanding the spectrum, we also cast the existing results in a new light.

Demand allocation with latency cost functions

We address the exact resolution of a Mixed Integer Non Linear Programming model where resources can be activated in order to satisfy a demand (a covering constraint) while minimizing total cost. For each resource, there is a fixed activation cost and a variable cost, expressed by means of latency functions. We prove that this problem is $${\mathcal {N} \mathcal {P}}$$ -hard even for linear latency functions. A branch and bound algorithm is devised, having two important features. First, a dual bound (equal to that obtained by continuous relaxation) can be computed very efficiently at each node of the enumeration tree. Second, to break symmetries resulting in improved efficiency, the branching scheme is n-ary (instead of binary). These features lead to a successful comparison against two popular commercial and open-source solvers, CPLEX and Bonmin.

Tight Bounds for Selfish and Greedy Load Balancing

We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it chooses, among its permissible servers, to run its job on the server having the smallest latency given the assignments of the jobs of other clients to servers. In online load balancing, clients appear online and, when a client appears, it has to make an irrevocable decision and assign its job to one of its permissible servers. Here, we assume that the clients aim to optimize some global criterion but in an online fashion. A natural local optimization criterion that can be used by each client when making its decision is to assign its job to that server that gives the minimum increase of the global objective. This gives rise to greedy online solutions. The aim of this paper is to determine how much the quality of load balancing is affected by selfishness and greediness. We characterize almost completely the impact of selfishness and greediness in load balancing by presenting new and improved, tight or almost tight bounds on the price of anarchy of selfish load balancing as well as on the competitiveness of the greedy algorithm for online load balancing when the objective is to minimize the total latency of all clients on servers with linear latency functions. In addition, we prove a tight upper bound on the price of stability of linear congestion games.

Throughput optimal total order broadcast for cluster environments

Total order broadcast is a fundamental communication primitive that plays a central role in bringing cheap software-based high availability to a wide range of services. This article studies the practical performance of such a primitive on a cluster of homogeneous machines. We present LCR, the first throughput optimal uniform total order broadcast protocol. LCR is based on a ring topology. It only relies on point-to-point inter-process communication and has a linear latency with respect to the number of processes. LCR is also fair in the sense that each process has an equal opportunity of having its messages delivered by all processes. We benchmark a C implementation of LCR against Spread and JGroups, two of the most widely used group communication packages. LCR provides higher throughput than the alternatives, over a large number of scenarios.